The Caucasus region is famed for its rich and diverse arrays of languages and language families, often challenging European-centered views established in traditional linguistics. In this paper, we describe ongoing efforts to improve the coverage of Universal Morphologies for languages of the Caucasus region. The Universal Morphologies (UniMorph) are a recent community project aiming to complement the Universal Dependencies which focus on morphosyntax and syntax. We describe the development of UniMorph resources for Nakh-Daghestanian and Kartvelian languages as a well as for Classical Armenian, we discuss challenges that the complex morphology of these and related languages poses to the current design of UniMorph, and suggest possibilities to improve the applicability of UniMorph for languages of the Caucasus region in particular and for low resource languages in general. We also criticize the UniMorph TSV format for its limited expressiveness, and suggest to complement the existing UniMorph workflow with support for additional source formats on grounds of Linked Open Data technology.